Singularly perturbed reaction–diffusion equations in a circle with numerical applications

REFERENCES

• P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, New York, 1973.
   Google Scholar
• W. Eckhaus, Boundary layers in linear elliptic singular perturbations, SIAM Rev. 14 (1972), pp. 225–270. doi: 10.1137/1014030
   Web of Science ®Google Scholar
• G. Gie, Singular perturbation problems in a general smooth domain, Asymptot. Anal. 62 (2009), pp. 227–249.
   Web of Science ®Google Scholar
• G. Gie, M. Hamouda, and R. Temam, Boundary layers in smooth curvilinear domains: Parabolic problems, Discrete Contin. Dyn. Syst. 26 (2010), pp. 1213–1240.
   Web of Science ®Google Scholar
• J. Grasman, An elliptic singular perturbation problem with almost characteristic boundaries, J. Math. Anal. Appl. 46 (1974), pp. 438–446. doi: 10.1016/0022-247X(74)90251-0
   Web of Science ®Google Scholar
• M. Hamouda, C.-Y. Jung, and R. Temam, Singular perturbations and boundary layer analysis, Springer, in preparation.
   Google Scholar
• Y. Hong, C.-Y. Jung, and R. Temam, On the numerical approximations of stiff convection–diffusion equations in a circle, submitted for publication.
   Google Scholar
• T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1987.
   Google Scholar
• C.-Y. Jung, Numerical approximation of two-dimensional convection–diffusion equations with boundary layers, Numer. Methods Partial Differential Equations 21 (2005), pp. 623–648. doi: 10.1002/num.20054
   Web of Science ®Google Scholar
• C.-Y. Jung and R. Temam, On parabolic boundary layers for convection–diffusion equations in a channel: Analysis and numerical applications, J. Sci. Comput. 28 (2006), pp. 361–410. doi: 10.1007/s10915-006-9086-8
   Web of Science ®Google Scholar
• C.-Y. Jung and R. Temam, Finite volume approximation of one-dimensional stiff convection–diffusion equations, J. Sci. Comput. 41 (2009), pp. 384–410. doi: 10.1007/s10915-009-9304-2
   Web of Science ®Google Scholar
• C.-Y. Jung and R. Temam, Finite volume approximation of two-dimensional stiff problems, Int. J. Numer. Anal. Model. 7 (2010), pp. 462–476.
   Web of Science ®Google Scholar
• C.-Y. Jung and R. Temam, Convection–diffusion equations in a circle: The compatible case, J. Math. Pures Appl. 96 (2011), pp. 88–107.
   Web of Science ®Google Scholar
• C.-Y. Jung, M. Petcu, and R. Temam, Singular perturbation analysis on a homogeneous ocean circulation model, Anal. Appl. (Singap.) 9 (2011), pp. 275–313. doi: 10.1142/S0219530511001832
   Web of Science ®Google Scholar
• R.B. Kellogg and M. Stynes, Layers and corner singularities in singularly perturbed elliptic problems, BIT 48 (2008), pp. 309–314. doi: 10.1007/s10543-008-0173-0
   Web of Science ®Google Scholar
• K.R. Knaub and R.E. O'Malley, The motion of internal layers in singularly perturbed advection–diffusion–reaction equations, Stud. Appl. Math 112 (2004), pp. 1–15. doi: 10.1111/j.1467-9590.2004.01502.x
   Web of Science ®Google Scholar
• O.A. Oleinik and V.N. Samokhin, Mathematical Models in Boundary Layer Theory, Applied Mathematics and Mathematical Computation, Chapman and Hall, Boca Raton, FL, 1999.
   Google Scholar
• R.E. O'Malley, Naive singular perturbation theory. Special issue in memory of Richard Weiss, Math. Models Methods Appl. Sci. 11 (2001), pp. 119–131. doi: 10.1142/S0218202501000787
   Web of Science ®Google Scholar
• P.O. Persson and G. Strang, A simple mesh generator in MATLAB, SIAM Rev. 46 (2004), pp. 329–345. doi: 10.1137/S0036144503429121
   Web of Science ®Google Scholar
• H.-G. Roos, M. Stynes, and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin, 1996.
   Google Scholar
• S. Shih and R.B. Kellogg, Asymptotic analysis of a singular perturbation problem, SIAM J. Math. Anal. 18 (1987), pp. 1467–1511. doi: 10.1137/0518107
   Web of Science ®Google Scholar
• G. Strang and G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973.
   Google Scholar
• M. Stynes, Steady-state convection diffusion problems, Acta Numer. 14 (2005), pp. 445–508. doi: 10.1017/S0962492904000261
   Google Scholar
• C. Xenophontos and S.R. Fulton, Uniform approximation of singularly perturbed reaction–diffusion problems by the finite element method on a Shishkin mesh, Numer. Methods Partial Differential Equations 19 (2003), pp. 89–111. doi: 10.1002/num.10034
   Web of Science ®Google Scholar